404 



SCIENCE. 



[N. S. Vol. XVII. No. 428. 



ments. Thus with Hilbert let a special 

 deductive or, mathematical science be based 

 upon a finite number of symbols related by 

 a finite number of compatible postulates, 

 every proposition of the science being de- 

 ducible by a finite number of logical steps 

 from the postulates. The content of this 

 conception is far from absolute. It in- 

 volves what presuppositions as to general 

 logic? What is a finite number? In 

 what sense is a postulate— for example, 

 that any two distinct points determine a 

 line— a single postulate? What are the 

 permissible logical steps of deduction? 

 Would the usual syllogistic steps of formal 

 logic suffice ? Would they suffice even with 

 the aid of the principle of mathematical 

 induction, in which Poincare finds* the 

 essential synthetic element of mathematical 

 argumentation the basis of that generality 

 without which there would be no science? 

 In what sense is mathematical induction a 

 single logical step of deduction? 



One has then the feeling that the carry- 

 ing out in an absolute sense of the program 

 of the abstract mathematicians will be 

 found impossible. At the same time, one 

 recognizes the importance attaching to the 

 effort to do precisely this thing. The re- 

 quirement of I'igor tends toward essential 

 simplicity of procedure, as Hilbert has in- 

 sisted in his Paris address, and the remark 

 applies to this question of mathematical 

 logic and its abstract expression. 



Pu7~e and Applied Mathematics. — In the 

 ultimate analysis for any epoch, we have 

 general logic, the mathematical sciences,! 

 that is, all special formally and abstractly 

 deductive self-consistent sciences, and the 

 natural sciences, which are inductive and 

 informally deductive. While this classifi- 

 cation may be satisfactory as an ideal one, 



* ' Sur la natvire du raisonnement mathfima- 

 tique,' Bevue de Metaphysique et de Morale, vol. 2 

 (1894), pp. 371-384. 



t Of which none is at present known to exist. 



it fails to recognize the fact that in mathe- 

 matical research one by no means confines 

 himself to processes which are mathemat- 

 ical according to this definition; and if 

 this is true with respect to the research of 

 professional mathematicians, how much 

 more is it true with respect to the study, 

 which should throughout be conducted in 

 the spirit of research, on the part of stu- 

 dents of mathematics in the elementary 

 schools and colleges and universities. I 

 refer to the articles* of Poincare on the 

 role of intuition and logic in mathematical 

 research and education. 



It is apparent that this ideal classifica- 

 tion can be made by the devotee of science 

 only when he has reached a considerable 

 degree of scientific maturity, that perhaps 

 it would fail to appeal to non-mathematical 

 experts, and that it does not accord with 

 the definitions given by practical working 

 mathematicians. Indeed, the attitude of 

 practical mathematicians toward this whole 

 subject of abstract mathematics, and espe- 

 cially the symbolic form of abstract mathe- 

 matics, is not unlike that of the practical 

 physicist toward the whole subject of theo- 

 retic mathematics, and in turn not unlike 

 that of the practical engineer toward the 

 whole subject of theoretical physics and 

 mathematics. Furthermore, every one 

 iinderstands that many of the most impor- 

 tant advances of pure mathematics have 

 arisen in connection with investigations 

 originating in the domain of natiiral phe- 

 nomena. 



Practically then it would seem desirable 



* ' La logique et I'intuition dans la science 

 mathgmatique et dans I'enseignenient,' L'En- 

 seignement MatMmatique, vol. 1 (1899), pp. 

 157-162. 'Du role de i'intuition et de la logique 

 en matheniatiques,' Oompte Rendu du Deuxieme 

 Congres International des Math4m.atioiens, Paris 

 [1900], 1902, pp. 115-130. 'Sur les rapports de 

 I'analyse pure et de la physique mathgmatique,' 

 Conference, Zurich, 1897; Acta Matliematica, vol. 

 21, p. 238. 



